A Compressed Sampling Receiver Based on Modulated Wideband Converter and a Parameter Estimation Algorithm for Fractional Bandlimited LFM Signals

Abstract

A modified modulated wideband converter (MWC)-based compressed sampling receiver and a parameter estimation algorithm in discrete time fractional Fourier domain (DTFrFD) to intercept and estimate the fractional bandlimited LFM signals are suggested in this work. Our proposed compressed sampling receiver has some advantages compared to the original MWC-based compressed sampling receiver. Firstly, the proposed receiver works in DTFrFD which is more effective to intercept the fractional bandlimited signals, while the original MWC-based digital receiver works in discrete time Fourier domain (DTFD). Secondly, the cross-channel signal can get better separation by the proposed digital receiver than by original MWC-based receiver because that the nonzero part of the signal’s available spectrum is narrower in DTFrFD than in DTFD. Then, with the data acquired from the digital receiver, we propose a novel parameter estimation algorithm for the intercepted fractional bandlimited LFM signal based on OMP and the differentiation spectrum in DTFrFD. The initial frequency and final frequency of the intercepted fractional bandlimited LFM signal can be extracted successfully without two-dimensional search and reconstruction, and only two iterations are needed according the algorithm. Both theoretical analysis and simulation results demonstrate that the proposed algorithm bears a relatively low complexity and its estimation precision is not only higher than both DFrFT-based search algorithm and DFrFT-based reconstruction algorithm but also close to Cramer–Rao lower bound.

Appendix A

Spectrum distribution characteristics of LFM signal in SFrFD A monocomponent LFM signal is defined by:

where $f_0$ is the initial frequency. A is the amplitude of $x\left( t\right)$ which could berandom or fixed. $K_{\mathrm{lfm}}$ is the modulation rate. And the duration time of $x\left( t\right)$ is $\left[ -\frac{T_d}{2},\frac{T_d}{2}\right]$.

The SFrFT of $x\left( t\right)$ can be calculated by

According to (S2), ${\overline{X}}_\alpha \left( u\right)$, the SFrFT of $x\left( u\right)$, denotes the FT of another LFM signal whose modulation rate is $K'_{\mathrm{lfm}} =K_{\mathrm{lfm}}+\cot \alpha /2 \pi$, initial frequency is $f_0$, and the frequency interval is $\left[ f_0-K'_{\mathrm{lfm}}\frac{T_d}{2}, f_0+K'_{\mathrm{lfm}}\frac{T_d}{2}\right]$. So the bandwidth is $B'=K'_{\mathrm{lfm}}\cdot T_d$. It can be interpreted that the SFrFT rotates the time-frequency distribution curve of $x\left( t\right)$ in the clockwise direction with angle $\alpha$, so that the modulation rate and bandwidth changes.

Thus,

$$\begin{aligned} {\overline{X}}_\alpha \left( u\right)= & {} \frac{A}{\sqrt{2K'_{\mathrm{lfm}}}}\exp \left( -\frac{j\left( u-2\pi u_0\csc \alpha \right) ^2}{4\pi K'_{\mathrm{lfm}}}\right) \cdot \left\{ \left[ c\left( u_2\right) \right. \right. \\&\left. \left. + c\left( u_1\right) \right] + j\left[ s\left( u_2\right) + s\left( u_1\right) \right] \right\} \end{aligned}$$

where $c\left( u\right) = \int _{0}^{u} \cos \left( \frac{\pi }{2}x^2 \right) \, \mathrm{d}x$ and $s\left( u\right) = \int _{0}^{u} \sin \left( \frac{\pi }{2}x^2 \right) \, \mathrm{d}x$ is the Fresnel integral. And

where $u_0=f_0\sin \alpha$. So the amplitude spectrum of ${\overline{X}}_\alpha \left( u\right)$ is

and the phase spectrum is

Substituting $K'_{\mathrm{lfm}}=\frac{B'}{T}$ into Eq. (S5):

According to the property associated with Fresnel integral, when $B'T_d \gg 1$, i.e., $K'_{\mathrm{lfm}}T_d^2 \gg 1$, $c\left( u_2\right) \approx s\left( u_2\right) \approx \frac{1}{2}$, so

$$\begin{aligned} |{{\overline{X}}}_\alpha \left( u\right) |\approx & {} \frac{A}{\sqrt{K'_{\mathrm{lfm}}}}\cdot rect\left( \frac{u-2\pi u_0\csc \alpha }{B'}\right) \\= & {} \frac{A}{\sqrt{K_{\mathrm{lfm}}+\cot \alpha /2 \pi }}\cdot rect\left( \frac{u-2\pi u_0\csc \alpha }{B'}\right) ,\theta \left( u\right) \\\approx & {} - \frac{\pi \left( u-2\pi u_0\csc \alpha \right) ^2}{K'_{\mathrm{lfm}}}+\frac{\pi }{4}=- \frac{\pi \left( u-2\pi u_0\csc \alpha \right) ^2}{K_{\mathrm{lfm}}+\cot \alpha /2 \pi }+\frac{\pi }{4}. \end{aligned}$$

Therefore, when $\left( K_{\mathrm{lfm}}+\cot \alpha /2 \pi \right) T_d^2\gg 1$, $|{{\overline{X}}}_\alpha \left( u\right) |$ is approximately a rectangular spectrum.